1/α + 1/β + 1/γ + 1/δ equals which expression?

1/α + 1/β + 1/γ + 1/δ equals which expression?

• A. (αβγ + βγδ + αγδ)/αβγδ
• B. (αβγδ)/αβγδ
• C. (αβ + γδ)/αβγδ
• D. (αβγ + βγδ + αγδ + αβδ)/αβγδ

Answer: (D)

Adding fractions with a common denominator is the idea here. To sum 1/α + 1/β + 1/γ + 1/δ, use the common denominator αβγδ. Rewriting each term over that denominator gives: 1/α = βγδ/(αβγδ), 1/β = αγδ/(αβγδ), 1/γ = αβδ/(αβγδ), 1/δ = αβγ/(αβγδ). Add the numerators: βγδ + αγδ + αβδ + αβγ, which is the same as αβγ + αβδ + αγδ + βγδ. So the sum is (αβγ + αβδ + αγδ + βγδ) / αβγδ. This matches the expression that has all four triple-product terms in the numerator over the product in the denominator. Other options would miss one of the triple products, collapse to 1, or pair terms in a way that doesn’t represent the full sum.

Adding fractions with a common denominator is the idea here. To sum 1/α + 1/β + 1/γ + 1/δ, use the common denominator αβγδ. Rewriting each term over that denominator gives:

1/α = βγδ/(αβγδ), 1/β = αγδ/(αβγδ), 1/γ = αβδ/(αβγδ), 1/δ = αβγ/(αβγδ).

Add the numerators: βγδ + αγδ + αβδ + αβγ, which is the same as αβγ + αβδ + αγδ + βγδ. So the sum is (αβγ + αβδ + αγδ + βγδ) / αβγδ. This matches the expression that has all four triple-product terms in the numerator over the product in the denominator.

Other options would miss one of the triple products, collapse to 1, or pair terms in a way that doesn’t represent the full sum.